Mesoscopic disorder in double-well optical lattices
Mesoscopic disorder in double-well optical lattices
arXiv:1107.3694v1 [cond-mat.quant-gas] 19 Jul 2011
V.I. Yukalov1 and E.P. Yukalova2 1
Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia 2
Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna 141980, Russia
Abstract Double-well optical lattices are considered, each cite of which is formed by a doublewell potential. The lattice is assumed to be in an insulating state and order and disorder are defined with respect to the displacement of atoms inside the double-well potential. It is shown that in such lattices, in addition to purely ordered and disordered states, there can exist an intermediate mixed state, where, inside a generally ordered lattice, there appear disordered regions of mesoscopic size.
E-mail: [email protected]
1
1
Introduction
Cold atoms, loaded into optical lattices, form the systems that are unique with regard to the controllability of their properties (see the review articles [1-5] and references therein). One can vary the sorts of atoms as well as the lattice types and lattice spacing. For the given kind of atoms, one can vary their interactions by means of the Feshbach resonance techniques [5-7]. It is also possible to change the lattice properties by imposing external fields or shaking the lattice. This is why, one is able to create a variety of different system states in the lattice [1-5]. A special type of lattices are the double-well lattices, whose experimental realization has become possible in recent years [8-12]. In such lattices, each lattice site is formed by a doublewell potential, which makes it feasible to create novel states that would be impossible in the standard single-well lattices. The phase diagram of bosons in double-well lattices has been studied [13,14], with the emphasis on the boundary of the insulator-superfluid transition. Double-well lattices in an insulator state have been analyzed [15-17], with the emphasis on collective excitations in these lattices [15], the order-disorder transition [15], dynamics of atomic loading in the lattice [15,16], and the feasibility of regulating atomic imbalance by varying the double-well potential shape [17]. The possibility of transferring atoms between the wells of a double-well potential by means of the stimulated Raman adiabatic passage scheme has been suggested [18]. In the present paper, the possibility of creating another state in a double-well optical lattice is advanced. We consider an insulating double-well lattice, with a single atom per cite, which is a setup that would be typical for the Mott insulator in a single-well lattice. Because of the insulating state and the unity filling factor, the atom statistics are not crucial; the atoms can be bosons as well as fermions. In such a system, there can exist an ordered state, when all atoms are unidirectionally shifted into one of the wells of each of the doublewell potentials [15]. Such a state is schematically shown in Fig. 1. It may also happen a disordered state, when atoms are randomly located in one of the wells of the doublewell potential [15]. We analyze under what conditions there could arise an intermediate mixed state, when inside a generally ordered lattice there could appear mesoscopic regions of disordered states, as is illustrated in Fig. 2. The term mesoscopic refers to the characteristic size lmes of the disordered regions, as compared to the lattice spacing a and to the total system size L. Of course, the disordered regions can be of different sizes and shapes, but it is always possible to define their characteristic, or average, size lmes . This size is mesoscopic in the sense of being intermediate between the lattice spacing and system sizes: a ≪ lmes ≪ L . We show that there exist such system parameters, when the intermediate mixed state of a lattice with mesoscopic disorder can arise. Since optical lattices allow for extremely wide possibilities in varying the system parameters, the suggested mixed state can be realized experimentally. And, as far as the properties of the mixed state can be regulated, it can be used for a variety of applications.
2
2
Random Region Configurations
In order to understand whether and when the mixed state could appear, it is necessary to take into account its possible existence, so that the probability of this state could be defined self-consistently. A system, formed by subregions corresponding to different thermodynamic states, having different orders, can be described by using the local-equilibrium Gibbs ensembles [19,20]. The regions of competing thermodynamics phases can arise in a self-organized way in arbitrary spatial locations, that is, their locations are randomly distributed in space. Therefore, it is necessary to average over these random configurations. Such a procedure has been suggested in Refs. [21-27] and reviewed in Ref. [28]. All mathematical details of this approach has been thoroughly described in these references. To better understand the following, in this section, we briefly delineate the main points of the approach as applied to a lattice system. Let us consider the lattice Z = {ai : i = 1, 2, . . . , NL }
(1)
composed of NL lattice cites characterized by the vectors ai . Suppose that there can exist two different thermodynamic states corresponding to an ordered and disordered phases. Meanwhile, it is not essential what kind of order is assumed. But for what follows, we keep in mind that each lattice cite is formed by a double-well potential and that the order is understood in the sense of the existence of an average atomic imbalance, as is explained in the Introduction. When a part of the lattice, say Z1 , is filled by an ordered state and a part Z2 , by a disordered one, the whole lattice is represented as the union \ [ Z = Z1 Z2 Z1 Z2 = 0 , (2) in which
NL = N1 + N2
(Nν ≡ mes Zν ) .
This is what is called the orthogonal covering. Each manifold covering can be characterized [29] by the manifold characteristic functions, or manifold indicators 1 , ai ∈ Zν , ξiν = (3) 0 , ai 6∈ Zν , describing the lattice separation into sublattices. The collection of the given manifold indicators composes the covering configuration ξ ≡ {ξiν : i = 1, 2, . . . , Nν ; ν = 1, 2} .
(4)
Each thermodynamic state is characterized by the corresponding space of typical microstates Hν that is a weighted Hilbert space, the complete metric space with a weighted norm induced by the inner product [28]. The latter implies that, if {ϕn } is an orthonormal basis in Hν and f ∈ Hν , so that the inner product (ϕ, f ) be given, then the weighted norm of f over Hν is s X ||f ||ν ≡ pnν |(ϕn , f )|2 , n
3
with the set {pnν } composing a probability measure: X 0 ≤ pnν ≤ 1 , pnν = 1 . n
The statistical average, with a statistical operator ρˆν , of an operator Aˆ on Hν is defined as X ˆ ˆ ˆ pnν ϕn , ρˆν Aϕn . hAiν = TrHν ρˆν A = n
The ordered and disordered states are distinguished as follows. There should exist an order operator ηˆ such that the averages ην ≡ hˆ η iν
(5)
define the order parameters for the corresponding ordered (η1 6= 0) and disordered (η2 = 0) states. For each fixed configuration ξ, the Hamiltonian energy operator, containing the singleatom terms Hi and the binary interaction terms Hij , possesses the representations X X ˆ ν (ξ) = H Hi ξiν + Hij ξiν ξjν (6) i6=j
i
defined on the related spaces Hν . The system as a whole is described by the Hamiltonian M ˆ 2 (ξ) ˆ 1 (ξ) H (7) H(ξ) = H
defined on the mixture space
M = H1
O
H2 .
(8)
The regions of disorder can arise in arbitrary locations and have different shapes. Therefore, it is necessary to invoke an averaging procedure over the random configuration set (4). Denoting the stochastic averaging by the double angle brackets >, one has the free energy F = −T lnhhTre−βH(ξ) ii , (9) where the trace is over space (8) and β = 1/T is inverse temperature. This corresponds to annealed disorder, since the disorder regions are not fixed in space by an external force, but ˆ appear in the system in a self-organized way. The stochastic averaging of an operator A(ξ) over the random configurations ξ can be rigorously defined [28] as a functional integration over the manifold indicators, Z ˆ ˆ Dξ. hhA(ξ)ii = A(ξ) (10)
Physically, the procedure of averaging over the random configurations is based on the existence of different spatial scales related to atoms (microscopic scale) and to the regions of disorder (mesoscopic scale). Different spatial scales also are often connected with different temporal scales of slow and fast motion [30-33]. Technically, the averaging over configurations reminds the renormalization procedure [34,35]. All exact mathematical details of the configuration averaging is given in the review article [28]. 4
Introducing the effective renormalized Hamiltonian e ≡ −T lnhhe−βH(ξ) ii H
(11)
makes it possible to rewrite the free energy (9) as
e
F = −T ln Tre−β H .
(12)
Thus, the basic point is to find the effective Hamiltonian (11). For the Hamiltonian representation (6), there has been proved [27,28] the following statement. Theorem. The effective Hamiltonian (11), where H(ξ) is given by the sum (7) of the Hamiltonian representations (6), takes the form M e =H ˆ1 ˆ2 , H H (13)
in which
ˆ ν = wν H
X
Hi + wν2
i
X
Hij ,
(14)
i6=j
with the weights wν being the minimizers of the free energy (12), under the conditions w1 + w2 = 1 ,
0 ≤ wν ≤ 1 .
(15)
The Hamiltonian form (13) is similar to the channel Hamiltonian representation in scattering theory [36,37].
3
Double-Well Insulating Lattice
The detailed derivation of the Hamiltonian for a double-well insulating lattice has been done in Ref. [15]. It is convenient to resort to pseudospin representation. To remind the reader the physical meaning of the pseudospin operators to be used in what follows, let us recall how they are constructed. We start with the operators of destruction and creation, cnj and c†nj , associated with the n-th energy level En of an atom in the j-th lattice cite. Considering an insulating lattice implies that atomic transitions between different lattice cites are suppressed, so that c†mi cnj = δij c†mi cni .
(16)
A lattice with the unity filling factor is assumed, when the number of atoms N coincides with the number of lattice cites NL . Then, limiting the consideration by two lowest energy levels of an atom in a double-well potential, one has the unipolarity condition c†1j c1j + c†2j c2j = 1 ,
cnj cnj = 0 .
The pseudospin operators are introduced through the transformations c†1j c1j =
1 + Sjx , 2
c†2j c2j = 5
1 − Sjx , 2
(17)
c†1j c2j = Sjz − iSjy ,
c†2j c1j = Sjz + iSjy .
(18)
The pseudospin operators Sjα satisfy the standard spin commutation relations for both the cases, when the operators cnj are either bosonic or fermionic. From the atomic operators, associated with the energy levels, one can pass to the operators, associated with the left and right positions of an atom in a double-well potential, by means of the transformations 1 c2j = √ (cjL − cjR ) . 2
1 c1j = √ (cjL + cjR ) , 2
(19)
As a result, the pseudospin operators take the form i † 1 † cjL cjR + c†jR cjL , Sjy = − cjL cjR − c†jR cjL , Sjx = 2 2 1 † † z Sj = c cjL − cjR cjR . (20) 2 jL The latter representation makes it clear the meaning of the pseudospin operators, showing that Sjx describes the atomic tunneling between the wells of a double-well potential in the j-th lattice cite, Sjy corresponds to the Josephson current between the wells, and the operator Sjz defines the atomic imbalance between the left and right wells in the j-the lattice cite. Employing these pseudospin operators yields [15] the single-atom energy part Hi = E0 − ΩSix ,
(21)
in which E0 = (E1 + E2 )/2 and Ω ≈ E2 − E1 is the interwell tunneling frequency. The binary interactions enter the part Hij =
1 Aij + Bij Six Sjx − Iij Siz Sjz , 2
(22)
with Aij being the direct interaction between atoms in the i-th and j-th cites, Bij , the interactions of atoms in the process of their tunneling between the wells, and Iij , the interactions of atoms shifted in one of the wells. All these quantities are explicitly expressed through the corresponding matrix elements of atomic interaction potentials over Wannier functions, as is done in Ref. [15]. For creating different states, it is important to have the possibility of varying atomic interactions in a wide range. In this regard, trapped atoms in optical lattices provide a unique opportunity. First of all, nowadays one is able to trap degenerate atomic clouds of a variety of atomic species possessing different scattering lengths. These are the whole series of bosonic atoms, 1 H, 4 He, 7 Li, 23 Na, 39 K, 41 K, 52 Cr, 85 Rb, 87 Rb, 133 Cs, 170 Yb, 174 Yb (see the summary in Refs. [1-5,38-44] and references therein) and also 40 Ca [45], 84 Sr [46,47], and 88 Sr [48]. There are experiments with several degenerate trapped Fermi gases, 3 He, 6 Li, 40 K, and 173 Yb (see review articles [49-51]). It is also possible to create different atomic molecules, made either of bosons, 23 Na2 , 85 Rb2 , 87 Rb2 , 133 Cs2 , or fermions, 6 Li2 , 40 K2 , as well as heteronuclear molecules, such as Bose-Fermi molecules 87 Rb-40 K or Bose-Bose molecules 85 Rb-87 Rb and 41 K-87 Rb [52]. The strength of atomic interactions can be varied in a very wide range by means of the Feshbach resonance techniques [6,7,38,52,53]. Finally, there are atoms with long-range dipolar interactions, such as 52 Cr possessing large magnetic moments [54,55], 6
and also Rydberg atoms [56] and polar molecules [57]. Moreover, in optical lattices, it is admissible to regulate the shape of the double-well potential, varying by this the tunneling frequency [15,58]. Therefore, the system parameters can really be varied in the range of several orders of magnitude. Employing the pseudospin representation, discussed above, for the effective Hamiltonian (14), we obtain 2 X X X ˆ ν = wν NE0 + wν H Bij Six Sjx − Iij Siz Sjz . Aij − wν Ω Sjx + wν2 2 i6=j j i6=j
(23)
To some extent, this Hamiltonian is analogous to the effective Hamiltonian describing solids with pores and cracks [59], in which the latter are the mesoscopic regions of disorder inside a regular solid matrix. For the insulating double-well lattice, the role of the order operator is played by the pseudospin operator Sjz , whose average gives the mean atomic imbalance. According to Sec. I, there are two kinds of averages, one defining the ordered state and another, disordered ˆ these two types of the averages are state. For an operator A, ˆν= hAi
ˆν) TrHν Aˆ exp(−β H , ˆν) TrHν exp(−β H
(24)
where ν = 1, 2. For the order operator, one has hSjz i1 6= 0 ,
hSjz i2 = 0 .
(25)
This means that the ordered state is characterized by a nonzero order parameter, while the disordered state, by the zero order parameter.
4
Mixture Order Parameters
It is convenient to define the reduced transverse averages yν ≡ 2hSiy iν ,
xν ≡ 2hSix iν ,
(26)
and the normalized order parameter sν ≡ 2hSiz iν .
(27)
To study the behavior of these averages under varying system parameters, let us resort to the mean-field approximation. Then, by using the standard mean-field techniques (see, e.g., [60]), for the free energy (12), we find Λν e F = F1 + F2 , Fν = Eν − NT ln 2 cosh , (28) 2T where the first term in Fν is
2 eν ≡ wν E0 N + wν A − Bx2 + Is2 N E ν ν 2
7
(29)
and in the second term, the notations p Λν ≡ wν Ω2ν + wν2 I 2 s2ν ,
Ων ≡ Ω − wν Bxν
(30)
are used. The parameters A, B, and I are given by the averages A≡
1 X Aij , N i6=j
B≡
1 X Bij , N i6=j
I≡
1 X Iij . N i6=j
For the mean tunneling intensity, defined in Eq. (26), we get Ων Λν xν = wν tanh . Λν 2T
(31)
(32)
The average Josephson current is zero, yν = 0 ,
(33)
as it should be in equilibrium. And for the order parameter (27), we find I Λν 2 sν = w ν sν tanh . Λν 2T
(34)
Let us introduce the dimensionless tunneling frequency and the tunneling interaction parameter, respectively, B Ω , b≡ . (35) ω≡ I +B I +B Also, let us define the dimensionless quantities ων ≡
Ων = ω − bwν xν , I +B
hν ≡
p Λν = wν ων2 + (1 − b)2 wν2 s2ν , I +B
(36)
corresponding to Eqs. (30). In what follows, let us measure temperature in units of I + B. Then the tunneling intensity (32) reads as ων hν xν = wν (37) tanh hν 2T and the order parameter (34) becomes sν =
wν2 sν
1−b tanh hν
hν 2T
.
(38)
In view of the definition of the ordered and disordered states, symbolized in conditions (25), we have s1 6= 0 , s2 = 0 . (39) Therefore for the ordered state, from Eqs. (38) and (39), we get the tunneling intensity x1 = 8
ω , w1
(40)
while the mean atomic imbalance s1 is defined by the equation h1 2 h1 = w1 (1 − b) tanh . 2T
(41)
For the disordered state, for which the mean imbalance is zero, s2 = 0, the tunneling intensity is w ω 2 2 x2 = tanh . (42) 2T In addition to the order parameters s1 and s2 , the mixed system is characterized by the quantities w1 and w2 , playing the role of the geometric weights of the corresponding states. These weights, according to the theorem of Sec. 2, are defined as the minimizers of the free energy (12), under the normalization conditions (15). This minimization yields 2u − ω1 x1 + ω2 x2 − (1 − b)s21 w2 = , 4u − (1 − b)s21
2u + ω1 x1 − ω2 x2 w1 = , 4u − (1 − b)s21
(43)
where the notation
A I +B 2 2 is used. The inequality d F/dw1 > 0 results in the condition 4u + b x21 + x22 − (1 − b)s21 > 0 . u≡
(44)
(45)
The thermodynamics of the mixed system is described by the set of Eqs. (39) to (43).
5
Quantum Phase Transition
Setting T = 0 in the above equations gives the tunneling intensities x1 = and the order parameters s1 =
s
ω , w1
1−
x2 = 1
ω2 , w12
(46)
s2 = 0 .
(47)
2u + b + ω − 1 . 4u + 2b − 1
(48)
The state weights (43) become w1 =
2u + b − ω , 4u + 2b − 1
w2 =
At zero temperature, the free energy coincides with the internal energy: e F = hHi
(T = 0) .
(49)
It is convenient to introduce the dimensionless internal energy E(w1 ) ≡
e hHi , N(I + B) 9
(50)
for which we find E(w1 ) = e0 + where
w1 w2 1 2u + b − 2ω − ω 2 − (2u + b − ω) + 1 (4u + 2b − 1) , 4 2 4
E0 . I +B The minimization procedure now implies that e0 ≡
∂ 2 E(w1 ) >0. ∂w12
∂E(w1 ) =0, ∂w1
(51)
(52)
(53)
The first of these equations leads to Eqs. (48), while the second gives the inequality 4u + 2b − 1 > 0 .
(54)
By its definition as a matrix element over Wannier functions [15], the interaction strength B is assumed to be positive and less than I. Then, in view of definition (35), one has 0 u0 . Hence, the mixed state exists only if u > uc . In this way, the quantum phase transition occurs at u = uc , given by Eq. (65), when w1 = 1. For u > uc , the mixed state is more stable, while for u < uc , the pure ordered state becomes preferable. When u → uc , then w1 → 1. And if u → ∞, then w1 → 1/2. The parameter u, defined in Eq. (44), has the meaning of the disorder parameter. For sufficiently large u > uc , the regions of mesoscopic disorder appear in the system, making this mixed state preferable as compared to the pure ordered state. uc ≡ u0 +
6
Temperature Phase Transition
In the pure system without mesoscopic disorder, when w1 ≡ 1, the order parameter s1 , describing the mean imbalance between the wells of a double-well potential, tends to zero at the temperature (1 − b)ω Tc = . (66) 2arctanh(ω) But, if the system is mixed, such that w1 is described by Eq. (48), then the transition temperature, at which s1 → 0, is Tc∗ =
(1 − b)ω , 4arctanh(2ω)
(67)
which is lower than temperature (66). Moreover, the situation for the mixed lattice is not as simple. The phase transition, at rising temperature, can happen not as a second-order 11
transition at Tc∗ but as a first-order transition at some intermediate temperature T0 between temperatures (66) and (67), provided that at this temperature T0 the mixed state becomes less favorable as compared to the disordered state. To study the behavior of the mixed state at finite temperatures, we take into account that the tunneling parameters ω and b are small, in agreement with inequalities (55) and (57). Hence, for numerical calculations, these parameters can be neglected. Then Eqs. (66) and (67) reduce to 1 1 Tc = , Tc∗ = . (68) 2 8 For convenience, we define the weights as w1 ≡ w ,
w2 = 1 − w .
And we solve numerically the equations for the order parameters 2 2u w s1 . , w= s1 = tanh 2T 4u − s21
(69)
(70)
These equations, generally, can possess several branches of solutions. Among them, we have to choose that one corresponding to the smaller free energy. Also, we need to compare the free energy of the mixed state with that of the pure system, for which w ≡ 1 and the order parameter is given by the equation s 1 (w = 1) . (71) s1 = tanh 2T
In addition, we have to consider the disordered state, for which s1 = 0 and w → 1/2. Comparing the free energies of all possible solutions, we select that solution whose free energy is minimal, which corresponds to the most stable system. In Figs. 3 and 4, we show the behavior of the stable branches of solutions as functions of temperature for different disorder parameters u. Figure 5 presents the dependence on the disorder parameter u of the order-disorder transition temperature T0 . The overall picture is as follows. (i) Nonpositive disorder parameter: u≤0.
(72)
Mesoscopic disorder does not appear at all. For temperatures below the critical Tc , the pure ordered state is stable, w=1 (0 ≤ T ≤ Tc ) . (73)
At Tc , the second-order phase transition takes place from the ordered state, where s1 6= 0, to the disordered state, where s1 = 0. (ii) In the region of the disorder parameter 0 < u ≤ un ,
(74)
where un ≈ 0.45, the pure ordered state remains till the temperature T0 , w=1
(0 ≤ T ≤ T0 ) , 12
(75)
where there happens a first-order phase transition to the disordered state. The transition temperature is in the interval Tc∗ < T0 < Tc , (76) depending on the value of u, as is shown in Fig. 5. (iii) In the region of the parameter un < u
arXiv:1107.3694v1 [cond-mat.quant-gas] 19 Jul 2011
V.I. Yukalov1 and E.P. Yukalova2 1
Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia 2
Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna 141980, Russia
Abstract Double-well optical lattices are considered, each cite of which is formed by a doublewell potential. The lattice is assumed to be in an insulating state and order and disorder are defined with respect to the displacement of atoms inside the double-well potential. It is shown that in such lattices, in addition to purely ordered and disordered states, there can exist an intermediate mixed state, where, inside a generally ordered lattice, there appear disordered regions of mesoscopic size.
E-mail: [email protected]
1
1
Introduction
Cold atoms, loaded into optical lattices, form the systems that are unique with regard to the controllability of their properties (see the review articles [1-5] and references therein). One can vary the sorts of atoms as well as the lattice types and lattice spacing. For the given kind of atoms, one can vary their interactions by means of the Feshbach resonance techniques [5-7]. It is also possible to change the lattice properties by imposing external fields or shaking the lattice. This is why, one is able to create a variety of different system states in the lattice [1-5]. A special type of lattices are the double-well lattices, whose experimental realization has become possible in recent years [8-12]. In such lattices, each lattice site is formed by a doublewell potential, which makes it feasible to create novel states that would be impossible in the standard single-well lattices. The phase diagram of bosons in double-well lattices has been studied [13,14], with the emphasis on the boundary of the insulator-superfluid transition. Double-well lattices in an insulator state have been analyzed [15-17], with the emphasis on collective excitations in these lattices [15], the order-disorder transition [15], dynamics of atomic loading in the lattice [15,16], and the feasibility of regulating atomic imbalance by varying the double-well potential shape [17]. The possibility of transferring atoms between the wells of a double-well potential by means of the stimulated Raman adiabatic passage scheme has been suggested [18]. In the present paper, the possibility of creating another state in a double-well optical lattice is advanced. We consider an insulating double-well lattice, with a single atom per cite, which is a setup that would be typical for the Mott insulator in a single-well lattice. Because of the insulating state and the unity filling factor, the atom statistics are not crucial; the atoms can be bosons as well as fermions. In such a system, there can exist an ordered state, when all atoms are unidirectionally shifted into one of the wells of each of the doublewell potentials [15]. Such a state is schematically shown in Fig. 1. It may also happen a disordered state, when atoms are randomly located in one of the wells of the doublewell potential [15]. We analyze under what conditions there could arise an intermediate mixed state, when inside a generally ordered lattice there could appear mesoscopic regions of disordered states, as is illustrated in Fig. 2. The term mesoscopic refers to the characteristic size lmes of the disordered regions, as compared to the lattice spacing a and to the total system size L. Of course, the disordered regions can be of different sizes and shapes, but it is always possible to define their characteristic, or average, size lmes . This size is mesoscopic in the sense of being intermediate between the lattice spacing and system sizes: a ≪ lmes ≪ L . We show that there exist such system parameters, when the intermediate mixed state of a lattice with mesoscopic disorder can arise. Since optical lattices allow for extremely wide possibilities in varying the system parameters, the suggested mixed state can be realized experimentally. And, as far as the properties of the mixed state can be regulated, it can be used for a variety of applications.
2
2
Random Region Configurations
In order to understand whether and when the mixed state could appear, it is necessary to take into account its possible existence, so that the probability of this state could be defined self-consistently. A system, formed by subregions corresponding to different thermodynamic states, having different orders, can be described by using the local-equilibrium Gibbs ensembles [19,20]. The regions of competing thermodynamics phases can arise in a self-organized way in arbitrary spatial locations, that is, their locations are randomly distributed in space. Therefore, it is necessary to average over these random configurations. Such a procedure has been suggested in Refs. [21-27] and reviewed in Ref. [28]. All mathematical details of this approach has been thoroughly described in these references. To better understand the following, in this section, we briefly delineate the main points of the approach as applied to a lattice system. Let us consider the lattice Z = {ai : i = 1, 2, . . . , NL }
(1)
composed of NL lattice cites characterized by the vectors ai . Suppose that there can exist two different thermodynamic states corresponding to an ordered and disordered phases. Meanwhile, it is not essential what kind of order is assumed. But for what follows, we keep in mind that each lattice cite is formed by a double-well potential and that the order is understood in the sense of the existence of an average atomic imbalance, as is explained in the Introduction. When a part of the lattice, say Z1 , is filled by an ordered state and a part Z2 , by a disordered one, the whole lattice is represented as the union \ [ Z = Z1 Z2 Z1 Z2 = 0 , (2) in which
NL = N1 + N2
(Nν ≡ mes Zν ) .
This is what is called the orthogonal covering. Each manifold covering can be characterized [29] by the manifold characteristic functions, or manifold indicators 1 , ai ∈ Zν , ξiν = (3) 0 , ai 6∈ Zν , describing the lattice separation into sublattices. The collection of the given manifold indicators composes the covering configuration ξ ≡ {ξiν : i = 1, 2, . . . , Nν ; ν = 1, 2} .
(4)
Each thermodynamic state is characterized by the corresponding space of typical microstates Hν that is a weighted Hilbert space, the complete metric space with a weighted norm induced by the inner product [28]. The latter implies that, if {ϕn } is an orthonormal basis in Hν and f ∈ Hν , so that the inner product (ϕ, f ) be given, then the weighted norm of f over Hν is s X ||f ||ν ≡ pnν |(ϕn , f )|2 , n
3
with the set {pnν } composing a probability measure: X 0 ≤ pnν ≤ 1 , pnν = 1 . n
The statistical average, with a statistical operator ρˆν , of an operator Aˆ on Hν is defined as X ˆ ˆ ˆ pnν ϕn , ρˆν Aϕn . hAiν = TrHν ρˆν A = n
The ordered and disordered states are distinguished as follows. There should exist an order operator ηˆ such that the averages ην ≡ hˆ η iν
(5)
define the order parameters for the corresponding ordered (η1 6= 0) and disordered (η2 = 0) states. For each fixed configuration ξ, the Hamiltonian energy operator, containing the singleatom terms Hi and the binary interaction terms Hij , possesses the representations X X ˆ ν (ξ) = H Hi ξiν + Hij ξiν ξjν (6) i6=j
i
defined on the related spaces Hν . The system as a whole is described by the Hamiltonian M ˆ 2 (ξ) ˆ 1 (ξ) H (7) H(ξ) = H
defined on the mixture space
M = H1
O
H2 .
(8)
The regions of disorder can arise in arbitrary locations and have different shapes. Therefore, it is necessary to invoke an averaging procedure over the random configuration set (4). Denoting the stochastic averaging by the double angle brackets >, one has the free energy F = −T lnhhTre−βH(ξ) ii , (9) where the trace is over space (8) and β = 1/T is inverse temperature. This corresponds to annealed disorder, since the disorder regions are not fixed in space by an external force, but ˆ appear in the system in a self-organized way. The stochastic averaging of an operator A(ξ) over the random configurations ξ can be rigorously defined [28] as a functional integration over the manifold indicators, Z ˆ ˆ Dξ. hhA(ξ)ii = A(ξ) (10)
Physically, the procedure of averaging over the random configurations is based on the existence of different spatial scales related to atoms (microscopic scale) and to the regions of disorder (mesoscopic scale). Different spatial scales also are often connected with different temporal scales of slow and fast motion [30-33]. Technically, the averaging over configurations reminds the renormalization procedure [34,35]. All exact mathematical details of the configuration averaging is given in the review article [28]. 4
Introducing the effective renormalized Hamiltonian e ≡ −T lnhhe−βH(ξ) ii H
(11)
makes it possible to rewrite the free energy (9) as
e
F = −T ln Tre−β H .
(12)
Thus, the basic point is to find the effective Hamiltonian (11). For the Hamiltonian representation (6), there has been proved [27,28] the following statement. Theorem. The effective Hamiltonian (11), where H(ξ) is given by the sum (7) of the Hamiltonian representations (6), takes the form M e =H ˆ1 ˆ2 , H H (13)
in which
ˆ ν = wν H
X
Hi + wν2
i
X
Hij ,
(14)
i6=j
with the weights wν being the minimizers of the free energy (12), under the conditions w1 + w2 = 1 ,
0 ≤ wν ≤ 1 .
(15)
The Hamiltonian form (13) is similar to the channel Hamiltonian representation in scattering theory [36,37].
3
Double-Well Insulating Lattice
The detailed derivation of the Hamiltonian for a double-well insulating lattice has been done in Ref. [15]. It is convenient to resort to pseudospin representation. To remind the reader the physical meaning of the pseudospin operators to be used in what follows, let us recall how they are constructed. We start with the operators of destruction and creation, cnj and c†nj , associated with the n-th energy level En of an atom in the j-th lattice cite. Considering an insulating lattice implies that atomic transitions between different lattice cites are suppressed, so that c†mi cnj = δij c†mi cni .
(16)
A lattice with the unity filling factor is assumed, when the number of atoms N coincides with the number of lattice cites NL . Then, limiting the consideration by two lowest energy levels of an atom in a double-well potential, one has the unipolarity condition c†1j c1j + c†2j c2j = 1 ,
cnj cnj = 0 .
The pseudospin operators are introduced through the transformations c†1j c1j =
1 + Sjx , 2
c†2j c2j = 5
1 − Sjx , 2
(17)
c†1j c2j = Sjz − iSjy ,
c†2j c1j = Sjz + iSjy .
(18)
The pseudospin operators Sjα satisfy the standard spin commutation relations for both the cases, when the operators cnj are either bosonic or fermionic. From the atomic operators, associated with the energy levels, one can pass to the operators, associated with the left and right positions of an atom in a double-well potential, by means of the transformations 1 c2j = √ (cjL − cjR ) . 2
1 c1j = √ (cjL + cjR ) , 2
(19)
As a result, the pseudospin operators take the form i † 1 † cjL cjR + c†jR cjL , Sjy = − cjL cjR − c†jR cjL , Sjx = 2 2 1 † † z Sj = c cjL − cjR cjR . (20) 2 jL The latter representation makes it clear the meaning of the pseudospin operators, showing that Sjx describes the atomic tunneling between the wells of a double-well potential in the j-th lattice cite, Sjy corresponds to the Josephson current between the wells, and the operator Sjz defines the atomic imbalance between the left and right wells in the j-the lattice cite. Employing these pseudospin operators yields [15] the single-atom energy part Hi = E0 − ΩSix ,
(21)
in which E0 = (E1 + E2 )/2 and Ω ≈ E2 − E1 is the interwell tunneling frequency. The binary interactions enter the part Hij =
1 Aij + Bij Six Sjx − Iij Siz Sjz , 2
(22)
with Aij being the direct interaction between atoms in the i-th and j-th cites, Bij , the interactions of atoms in the process of their tunneling between the wells, and Iij , the interactions of atoms shifted in one of the wells. All these quantities are explicitly expressed through the corresponding matrix elements of atomic interaction potentials over Wannier functions, as is done in Ref. [15]. For creating different states, it is important to have the possibility of varying atomic interactions in a wide range. In this regard, trapped atoms in optical lattices provide a unique opportunity. First of all, nowadays one is able to trap degenerate atomic clouds of a variety of atomic species possessing different scattering lengths. These are the whole series of bosonic atoms, 1 H, 4 He, 7 Li, 23 Na, 39 K, 41 K, 52 Cr, 85 Rb, 87 Rb, 133 Cs, 170 Yb, 174 Yb (see the summary in Refs. [1-5,38-44] and references therein) and also 40 Ca [45], 84 Sr [46,47], and 88 Sr [48]. There are experiments with several degenerate trapped Fermi gases, 3 He, 6 Li, 40 K, and 173 Yb (see review articles [49-51]). It is also possible to create different atomic molecules, made either of bosons, 23 Na2 , 85 Rb2 , 87 Rb2 , 133 Cs2 , or fermions, 6 Li2 , 40 K2 , as well as heteronuclear molecules, such as Bose-Fermi molecules 87 Rb-40 K or Bose-Bose molecules 85 Rb-87 Rb and 41 K-87 Rb [52]. The strength of atomic interactions can be varied in a very wide range by means of the Feshbach resonance techniques [6,7,38,52,53]. Finally, there are atoms with long-range dipolar interactions, such as 52 Cr possessing large magnetic moments [54,55], 6
and also Rydberg atoms [56] and polar molecules [57]. Moreover, in optical lattices, it is admissible to regulate the shape of the double-well potential, varying by this the tunneling frequency [15,58]. Therefore, the system parameters can really be varied in the range of several orders of magnitude. Employing the pseudospin representation, discussed above, for the effective Hamiltonian (14), we obtain 2 X X X ˆ ν = wν NE0 + wν H Bij Six Sjx − Iij Siz Sjz . Aij − wν Ω Sjx + wν2 2 i6=j j i6=j
(23)
To some extent, this Hamiltonian is analogous to the effective Hamiltonian describing solids with pores and cracks [59], in which the latter are the mesoscopic regions of disorder inside a regular solid matrix. For the insulating double-well lattice, the role of the order operator is played by the pseudospin operator Sjz , whose average gives the mean atomic imbalance. According to Sec. I, there are two kinds of averages, one defining the ordered state and another, disordered ˆ these two types of the averages are state. For an operator A, ˆν= hAi
ˆν) TrHν Aˆ exp(−β H , ˆν) TrHν exp(−β H
(24)
where ν = 1, 2. For the order operator, one has hSjz i1 6= 0 ,
hSjz i2 = 0 .
(25)
This means that the ordered state is characterized by a nonzero order parameter, while the disordered state, by the zero order parameter.
4
Mixture Order Parameters
It is convenient to define the reduced transverse averages yν ≡ 2hSiy iν ,
xν ≡ 2hSix iν ,
(26)
and the normalized order parameter sν ≡ 2hSiz iν .
(27)
To study the behavior of these averages under varying system parameters, let us resort to the mean-field approximation. Then, by using the standard mean-field techniques (see, e.g., [60]), for the free energy (12), we find Λν e F = F1 + F2 , Fν = Eν − NT ln 2 cosh , (28) 2T where the first term in Fν is
2 eν ≡ wν E0 N + wν A − Bx2 + Is2 N E ν ν 2
7
(29)
and in the second term, the notations p Λν ≡ wν Ω2ν + wν2 I 2 s2ν ,
Ων ≡ Ω − wν Bxν
(30)
are used. The parameters A, B, and I are given by the averages A≡
1 X Aij , N i6=j
B≡
1 X Bij , N i6=j
I≡
1 X Iij . N i6=j
For the mean tunneling intensity, defined in Eq. (26), we get Ων Λν xν = wν tanh . Λν 2T
(31)
(32)
The average Josephson current is zero, yν = 0 ,
(33)
as it should be in equilibrium. And for the order parameter (27), we find I Λν 2 sν = w ν sν tanh . Λν 2T
(34)
Let us introduce the dimensionless tunneling frequency and the tunneling interaction parameter, respectively, B Ω , b≡ . (35) ω≡ I +B I +B Also, let us define the dimensionless quantities ων ≡
Ων = ω − bwν xν , I +B
hν ≡
p Λν = wν ων2 + (1 − b)2 wν2 s2ν , I +B
(36)
corresponding to Eqs. (30). In what follows, let us measure temperature in units of I + B. Then the tunneling intensity (32) reads as ων hν xν = wν (37) tanh hν 2T and the order parameter (34) becomes sν =
wν2 sν
1−b tanh hν
hν 2T
.
(38)
In view of the definition of the ordered and disordered states, symbolized in conditions (25), we have s1 6= 0 , s2 = 0 . (39) Therefore for the ordered state, from Eqs. (38) and (39), we get the tunneling intensity x1 = 8
ω , w1
(40)
while the mean atomic imbalance s1 is defined by the equation h1 2 h1 = w1 (1 − b) tanh . 2T
(41)
For the disordered state, for which the mean imbalance is zero, s2 = 0, the tunneling intensity is w ω 2 2 x2 = tanh . (42) 2T In addition to the order parameters s1 and s2 , the mixed system is characterized by the quantities w1 and w2 , playing the role of the geometric weights of the corresponding states. These weights, according to the theorem of Sec. 2, are defined as the minimizers of the free energy (12), under the normalization conditions (15). This minimization yields 2u − ω1 x1 + ω2 x2 − (1 − b)s21 w2 = , 4u − (1 − b)s21
2u + ω1 x1 − ω2 x2 w1 = , 4u − (1 − b)s21
(43)
where the notation
A I +B 2 2 is used. The inequality d F/dw1 > 0 results in the condition 4u + b x21 + x22 − (1 − b)s21 > 0 . u≡
(44)
(45)
The thermodynamics of the mixed system is described by the set of Eqs. (39) to (43).
5
Quantum Phase Transition
Setting T = 0 in the above equations gives the tunneling intensities x1 = and the order parameters s1 =
s
ω , w1
1−
x2 = 1
ω2 , w12
(46)
s2 = 0 .
(47)
2u + b + ω − 1 . 4u + 2b − 1
(48)
The state weights (43) become w1 =
2u + b − ω , 4u + 2b − 1
w2 =
At zero temperature, the free energy coincides with the internal energy: e F = hHi
(T = 0) .
(49)
It is convenient to introduce the dimensionless internal energy E(w1 ) ≡
e hHi , N(I + B) 9
(50)
for which we find E(w1 ) = e0 + where
w1 w2 1 2u + b − 2ω − ω 2 − (2u + b − ω) + 1 (4u + 2b − 1) , 4 2 4
E0 . I +B The minimization procedure now implies that e0 ≡
∂ 2 E(w1 ) >0. ∂w12
∂E(w1 ) =0, ∂w1
(51)
(52)
(53)
The first of these equations leads to Eqs. (48), while the second gives the inequality 4u + 2b − 1 > 0 .
(54)
By its definition as a matrix element over Wannier functions [15], the interaction strength B is assumed to be positive and less than I. Then, in view of definition (35), one has 0 u0 . Hence, the mixed state exists only if u > uc . In this way, the quantum phase transition occurs at u = uc , given by Eq. (65), when w1 = 1. For u > uc , the mixed state is more stable, while for u < uc , the pure ordered state becomes preferable. When u → uc , then w1 → 1. And if u → ∞, then w1 → 1/2. The parameter u, defined in Eq. (44), has the meaning of the disorder parameter. For sufficiently large u > uc , the regions of mesoscopic disorder appear in the system, making this mixed state preferable as compared to the pure ordered state. uc ≡ u0 +
6
Temperature Phase Transition
In the pure system without mesoscopic disorder, when w1 ≡ 1, the order parameter s1 , describing the mean imbalance between the wells of a double-well potential, tends to zero at the temperature (1 − b)ω Tc = . (66) 2arctanh(ω) But, if the system is mixed, such that w1 is described by Eq. (48), then the transition temperature, at which s1 → 0, is Tc∗ =
(1 − b)ω , 4arctanh(2ω)
(67)
which is lower than temperature (66). Moreover, the situation for the mixed lattice is not as simple. The phase transition, at rising temperature, can happen not as a second-order 11
transition at Tc∗ but as a first-order transition at some intermediate temperature T0 between temperatures (66) and (67), provided that at this temperature T0 the mixed state becomes less favorable as compared to the disordered state. To study the behavior of the mixed state at finite temperatures, we take into account that the tunneling parameters ω and b are small, in agreement with inequalities (55) and (57). Hence, for numerical calculations, these parameters can be neglected. Then Eqs. (66) and (67) reduce to 1 1 Tc = , Tc∗ = . (68) 2 8 For convenience, we define the weights as w1 ≡ w ,
w2 = 1 − w .
And we solve numerically the equations for the order parameters 2 2u w s1 . , w= s1 = tanh 2T 4u − s21
(69)
(70)
These equations, generally, can possess several branches of solutions. Among them, we have to choose that one corresponding to the smaller free energy. Also, we need to compare the free energy of the mixed state with that of the pure system, for which w ≡ 1 and the order parameter is given by the equation s 1 (w = 1) . (71) s1 = tanh 2T
In addition, we have to consider the disordered state, for which s1 = 0 and w → 1/2. Comparing the free energies of all possible solutions, we select that solution whose free energy is minimal, which corresponds to the most stable system. In Figs. 3 and 4, we show the behavior of the stable branches of solutions as functions of temperature for different disorder parameters u. Figure 5 presents the dependence on the disorder parameter u of the order-disorder transition temperature T0 . The overall picture is as follows. (i) Nonpositive disorder parameter: u≤0.
(72)
Mesoscopic disorder does not appear at all. For temperatures below the critical Tc , the pure ordered state is stable, w=1 (0 ≤ T ≤ Tc ) . (73)
At Tc , the second-order phase transition takes place from the ordered state, where s1 6= 0, to the disordered state, where s1 = 0. (ii) In the region of the disorder parameter 0 < u ≤ un ,
(74)
where un ≈ 0.45, the pure ordered state remains till the temperature T0 , w=1
(0 ≤ T ≤ T0 ) , 12
(75)
where there happens a first-order phase transition to the disordered state. The transition temperature is in the interval Tc∗ < T0 < Tc , (76) depending on the value of u, as is shown in Fig. 5. (iii) In the region of the parameter un < u
